严格上三角矩阵李代数上的积零导子

设$F$ 为域, $ngeq 3$, $bf{N}$$(n,mathbb{F})$ 为域$mathbb{F}$ 上所有$ntimes n$ 阶严格上三角矩阵构成的严格上三角矩阵李代数, 其李运算为$[x,y]=xy-yx$. $bf{N}$$(n, mathbb{F})$ 上一线性映射$varphi$ 称为积零导子,如果由$[x,y]=0, x,yin bf{N}$$(n,mathbb{F})$,总可推出 $[varphi(x), y]+[x,varphi(y)]=0$. 本文证明 $bf{N}$$(n,mathbb{F})$上一线性映射 $varphi$ 为积零导子当且仅当 $varphi$ 为$bf{N}$$(n,mathbb{F})$ 上内导子, 对角线导子, 极端导子, 中心导子和标量乘法的和.

Product Zero Derivations on Strictly Upper Triangular Matrix Lie Algebras

Let $mathbb{F}$ be a field, $ngeq 3$, ${bf N}(n,mathbb{F})$ the strictly upper triangular matrix Lie algebra consisting of the $ntimes n$ strictly upper triangular matrices and with the bracket operation $[x,y]=xy-yx$. A linear map $varphi$ on ${bf N}(n, mathbb{F})$ is said to be a product zero derivation if $[varphi(x), y]+[x, varphi(y)]=0$ whenever $[x,y]=0, x,yin {bf N}(n,mathbb{F})$. In this paper, we prove that a linear map on ${bf N}(n,mathbb{F})$ is a product zero derivation if and only if $varphi$ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on ${bf N}(n,mathbb{F})$.